Anion-exchange membrane fuel cells (AEMFCs) are a promising, next-generation fuel cell technology. AEMFCs require highly conductive and robust anion-exchange membranes (AEMs), which are challenging to develop due to the tradeoff between conductivity and water uptake. Here we report a method to prepare high-molecular-weight branched poly(aryl piperidinium) AEMs. We show that branching reduces water uptake, leading to improved dimensional stability. The optimized membrane, b-PTP-2.5, exhibits simultaneously high OH- conductivity (>145 mS cm(-1) at 80 degrees C), high mechanical strength and dimensional stability, good processability, and excellent alkaline stability (>1500 h) in 1 M KOH at 80 degrees C. AEMFCs based on b-PTP-2.5 reached peak power densities of 2.3 W cm(-2) in H-2-O-2 and 1.3 W cm(-2) in H-2-air at 80 degrees C. The AEMFCs can run stably under a constant current of 0.2 A cm(-2) over 500 h, during which the b-PTP-2.5 membrane remains stable.